from matplotlib import cm
import numpy as np
import matplotlib.pyplot as plt

eps = 0.25
L   = 500
N   = 8
x0  = 1.0

t = np.linspace(0, eps, L)

def F(t, x):
    return x**2

y = np.zeros((N+1, L))
y[0, :] = x0

for n in range(N):
    for k in range(L):
        integrando = F(t[:k+1], y[n, :k+1])
        y[n+1, k]  = x0 + np.trapezoid(integrando, t[:k+1])

def x_exata(t):
    return 1.0 / (1.0 - t)

x_ref = x_exata(t)

print(f"{'n':<4} {'trapézio':>12}")
print("-" * 30)
for n in range(N+1):
    erro_tp = np.max(np.abs(y[n] - x_ref))
    print(f"{n:<4} {erro_tp:>12.2e}")

fig, ax1 = plt.subplots(1, 1, figsize=(7, 5))

cores = cm.viridis(np.linspace(0, 1, N+1))

for n in range(1, N+1):
    ax1.plot(t, y[n], color=cores[n])
ax1.plot(t, x_ref, linestyle='--', color='red', linewidth=2, label='solução exata')

ax1.set_xlabel("$t$")
ax1.set_ylabel("$x$")
ax1.set_xlim(0, eps)
ax1.set_ylim(0, 2)
ax1.grid(True)

sm = plt.cm.ScalarMappable(cmap='viridis',
     norm=plt.Normalize(vmin=1, vmax=N))
plt.colorbar(sm, ax=ax1, label='iterado $n$')

ax1.legend()
plt.tight_layout()
plt.show()